The MABAC method using for supply chain management based on the bipolar complex fuzzy soft hamacher aggregation operators

Complex fuzzy sets and its accompanying theory although at its infancy, has proven to be superior to classical type-1 fuzzy sets, due its ability in representing time-periodic problem parameters and capturing the seasonality of the fuzziness that exists in the elements of a set. These are important characteristics that are pervasive in most real world problems. However, there are two major problems that are inherent in complex fuzzy sets: it lacks a sufficient parameterization tool and it does not have a mechanism to validate the values assigned to the membership functions of the elements in a set. To overcome these problems, we propose the notion of complex fuzzy soft expert sets which is a hybrid model of complex fuzzy sets and soft expert sets. This model incorporates the advantages of complex fuzzy sets and soft sets, besides having the added advantage of allowing the users to know the opinion of all the experts in a single model without the need for any additional cumbersome operations. As such, this model effectively improves the accuracy of representation of problem parameters that are periodic in nature, besides having a higher level of computational efficiency compared to similar models in literature.

Type-2 fuzzy sets allow us to incorporate the uncertainties about the membership functions into fuzzy sets, thereby overcoming a problem that is inherent in type-1 fuzzy sets, which does not allow for any uncertainty in assigning values to the membership functions. Complex fuzzy sets are type-1 fuzzy sets with complex-valued grades of membership and are characterized by an additional phase term which enables it to better represent and capture the time-periodic and seasonal aspects of fuzziness that are prevalent in many real world problems and time-series applications. However, similar to type-1 fuzzy sets, the membership functions of complex fuzzy sets are difficult to enumerate, as they are subject to individual preferences and bias. To overcome this problem, we propose the concept of interval-valued complex fuzzy soft sets which combines complex fuzzy sets with type-2 fuzzy sets and soft sets. This adaption of complex fuzzy sets assigns an interval-based membership to each element and adequate parameterization, which betters corresponds to the intuition of representing fuzzy data. Subsequently this paper is concerned with the concepts related to this model, verifying the algebraic properties and demonstrating the utility of this model.

Fuzzy set theory, rough set theory, and soft set theory are three effective mathematical tools for dealing with uncertainties and have many wide applications both in theory and practise. Meng et al. (2011) introduced the notion of soft fuzzy rough sets by combining fuzzy sets, rough sets, and soft sets all together. The aim of this paper is to study the parameter reduction of fuzzy soft sets based on soft fuzzy rough approximation operators. We propose some concepts and conditions for two fuzzy soft sets to generate the same lower soft fuzzy rough approximation operators and the same upper soft fuzzy rough approximation operators. The concept of reduct of a fuzzy soft set is introduced and the procedure to find a reduct for a fuzzy soft set is given. Furthermore, the concept of exclusion of a fuzzy soft set is introduced and the procedure to find an exclusion for a fuzzy soft set is given.

The objective of this article is to explore and generalize the notions of soft set and rough set along with spherical fuzzy set and to introduce the novel concept called spherical fuzzy soft rough set that is free from all those complications faced by many modern concepts like intuitionistic fuzzy soft rough set, Pythagorean fuzzy soft rough set, and q-rung orthopair fuzzy soft rough set. Since aggregation operators are the fundamental tools to translate the complete information into a distinct number, so some spherical fuzzy soft rough new average aggregation operators are introduced, such as spherical fuzzy soft rough weighted average, spherical fuzzy soft rough ordered weighted average, and spherical fuzzy soft rough hybrid average aggregation operators. Also, the basic characteristics of these introduced operators have been elaborated in detail. Furthermore, a multi-criteria decision-making (MCDM) technique has been developed and a descriptive example is given to support newly presented work. At the end of this article, a comparative study of the introduced technique has been established that shows how our work is more superior and efficient compared to the picture fuzzy soft set.

Prior studies in sustainable supply chain management (SSCM) fail to address collaboration in the supply chain and lean management in conjunction with economic benefits. Collaboration in the supply chain and lean management have to integrate into SSCM, which involves social, economic and environmental aspects. This study aims to establish a framework to understand the seafood industry in Vietnam and to enhance its performance. This framework proposes 5 aspects and 21 criteria, and these aspects involve qualitative information. Fuzzy set theory is applied to deal with the complexity of and uncertainty in the linguistic preferences. The fuzzy Delphi method is employed to test the reliability and validity attributes. A decision-making trial and evaluation laboratory is to assess the interrelationships among the attributes. Sensitivity analysis is adopted to address subjective robustness and objective perceptions. The results show that the aspects of collaboration in the supply chain and lean management drive economic benefits.

Emerging trends in soft set theory and related topics.

In this manuscript we demonstrate interval-valued bipolar complex fuzzy set (IVBCFS) and then interval-valued bipolar complex fuzzy soft set (IVBCFSS), as a generalization of fuzzy set, interval-valued fuzzy set, bipolar fuzzy set, complex fuzzy set and soft set. We also initiate operational laws and basic results and properties for IVBCFS and IVBCFSS. Further explanation is given for the basic algebraic operations like complement, extended union, extended intersection, restricted union, and restricted intersection, AND product and OR product for IVBCFSS. Moreover, we demonstrate some fundamental aggregation operators like IVBCFS average aggregation, IVBCFS geometric and as well as their properties. To emphasize the usefulness and application of the system, we also develop the decision-making method and joint instances of the IVBCFSS (set-ups 1 and 2). In order to describe the effectiveness and influence of the approaching novel work, this study uses a comparative analysis of the new creating concept with prevailing ideas.

Bipolar complex fuzzy set (BCFS) is a more advanced and powerful phenomenon as it consists of two-dimensional data with positive and negative impacts of an element. It can solve the data consisting of the positive and negative impacts of an element which is a bipolar fuzzy set (BFS). It also covers the two-dimensional complex data which is a complex fuzzy set (CFS). Due to these attributes, BFS and CFS are less useful in comparison with BCFS to capture vagueness, complexity, and ambiguity in the data. Furthermore, lower and upper approximations based on equivalency relations constitute another significant phenomenon known as rough set (RS). This structure is also more powerful in dealing with real-life dilemmas. Rather than comparing the RS and BCFS, we combine both phenomena to handle the complexity more powerfully to deal with such types of phenomena that are not handled by other structures. So, by combining both phenomena, we introduce a novel structure known to be bipolar complex fuzzy rough set (BCFRS) in this manuscript. After that, we define some important operations, some significant properties related to this structure, and some aggregation operators (AOs) to solve decision-making (DM) problems related to cyber security. We address a practical application of cyber security (C-S) in computing for the protection of critical data to demonstrate the usefulness of the multi-attribute DM(MADM) approach. Based on the various criteria and attributes given by the experts, we find the best and better alternative to the C-S by applying the MADM approach. We get the A4 as the best and finest alternative by using bipolar complex fuzzy rough (BCFR) weighted arithmetic averaging (BCFRWAA), BCFR ordered weighted arithmetic averaging (BCFROWAA), and BCFR ordered weighted geometric averaging (BCFROWGA) operators. And, by using BCFR weighted geometric averaging (BCFRWGA), we get the A3 as the finest alternative. Lastly, to prove the superiority, validity, and generalization of our unique established theory, we give a detailed comparative study of our established work with several prevalent theories.

Complex fuzzy set, as an extension of classical fuzzy sets, could describe the fuzzy characters of things more detail and comprehensively and is very useful in dealing with vagueness and uncertainty of problems that include the periodic or recurring phenomena. Note that a complex fuzzy set is different from the fuzzy complex set introduced and discussed by many scholars, since the membership degree of a complex fuzzy set is a complex number with length less than or equal to 1 while a fuzzy complex set is a real number with membership degree less than or equal to 1, and the universe is the complex plane. As the mathematical theoretical basis of fuzzy mathematics, fuzzy set and its mapping, corresponding fuzzy complex set and its mapping have been investigated in depth because they integrate and cross the methods and results of classical real analysis and complex analysis. However, there is no comprehensive investigation on complex fuzzy set and its corresponding mathematical theory, even include decomposition theorems, extension principles and the basic operations of the complex fuzzy set. As is well known, the cut set of fuzzy sets is the bridge between fuzzy sets and classical sets, which plays a significant role in fuzzy sets and fuzzy systems. In this paper, the concept of (r, θ)-cut sets of complex fuzzy sets is proposed and their properties are discussed. Meanwhile, the decomposition theorems and the extension principles of complex fuzzy set based on (r, θ)-cut sets are deduced and corresponding properties are investigated. All these conclusions not only deeply enrich the fundamental theory of complex fuzzy set, but also provide a powerful tool to investigate complex fuzzy set. Finally, an example application of signal detection demonstrates the utility of the (r, θ)-cut sets of complex fuzzy sets in practice.

Complex fuzzy (CF) sets (CFSs) have a significant role in modelling the problems involving two-dimensional information. Recently, the extensions of CFSs have gained the attention of researchers studying decision-making methods. The complex T-spherical fuzzy set (CTSFS) is an extension of the CFSs introduced in the last times. In this paper, we introduce the Dombi operations on CTSFSs. Based on Dombi operators, we define some aggregation operators, including complex T-spherical Dombi fuzzy weighted arithmetic averaging (CTSDFWAA) operator, complex T-spherical Dombi fuzzy weighted geometric averaging (CTSDFWGA) operator, complex T-spherical Dombi fuzzy ordered weighted arithmetic averaging (CTSDFOWAA) operator, complex T-spherical Dombi fuzzy ordered weighted geometric averaging (CTSDFOWGA) operator, and we obtain some of their properties. In addition, we develop a multi-criteria decision-making (MCDM) method under the CTSF environment and present an algorithm for the proposed method. To show the process of the proposed method, we present an example related to diagnosing the COVID-19. Besides this, we present a sensitivity analysis to reveal the advantages and restrictions of our method.

Analysis of renewable energy resources based on frank power aggregation operators and EDAS method for circular bipolar complex fuzzy uncertainty

Complex fuzzy N-soft set (CFN-SS) is an important technique to manage awkward and unreliable information in realistic decision-making problems. CFN-SS is a blend of two separate theories, called N-soft sets (N-SSs) and complex fuzzy sets (CFSs), which are the modified versions of soft sets (SSs) and fuzzy sets (FSs) to depict vague and uncertain information in daily life problems. In this manuscript, the novel concept of CFN-SS is explored and their fundamental laws are discussed. CFN-SS contains the grade of truth in the form of a complex number whose real and imaginary parts are limited to the unit interval. Besides, we examine some algebraic properties for CFN-SS like union, intersections and justify these properties with the help of some numerical examples. To examine the superiority and effectiveness of the proposed approaches, the special cases of the investigated approaches are also discussed. A decision-making procedure is developed by using the investigated ideas based on CFN-SSs. Further, some numerical examples are also illustrated with the help of explored ideas to find the reliability and effectiveness of the proposed approaches. Finally, the comparative analysis of the investigated ideas with some existing ideas is also demonstrated to prove the quality of the proposed works. The graphical expressions of the obtained results are also discussed.

In recent days researchers have tried to handle the maximum information and use those techniques and methods in which there is no chance of data loss or loss of information is minimum. The structure like fuzzy set and complex fussy set cannot discuss the upper and lower approximations. Moreover, we can observe that a fuzzy rough set cannot discuss the second dimension and in this case, there is a chance of data loss. To cover all these issues in previous ideas, the notion of a complex fuzzy rough set in Cartesian form is the demand of the day because this structure can discuss the second dimension as well as upper and lower approximations. For this purpose, in this manuscript, we have developed the theory of complex fuzzy relation and complex fuzzy rough set in Cartesian form. Moreover, we have initiated the fundamental laws for complex fuzzy rough numbers based on Frank t-norm and t-conorm. The fundamental tools that can convert the overall input into a single output are called aggregation operators (AOs). So based on the characteristics of AOs, we have defined the notion of complex fuzzy rough Frank average and complex fuzzy rough Frank geometric AOs. The utilization of the developed theory is necessary to show the importance and validity of the delivered approach. So based on developed notions, we have defined an algorithm for this purpose along with an illustrative example. We have utilized the introduced structure for the classification of AI tools for civil engineering. Moreover, the comparative analysis of the delivered approach shows the advancement of the introduced structure as compared to existing notions.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

The model of bipolar complex fuzzy linguistic set is a very famous and dominant principle to cope with vague and uncertain information. The bipolar complex fuzzy linguistic set contained the positive membership function, negative membership function, and linguistic variable, where the technique of fuzzy sets to bipolar fuzzy sets are the special cases of the bipolar complex fuzzy linguistic set. In this manuscript, we describe the model of Aczel-Alsina operational laws for bipolar complex fuzzy linguistic values based on Aczel-Alsina t-norm and Aczel-Alsina t-conorm. Additionally, we compute the Aczel-Alsina power aggregation operators based on bipolar complex fuzzy linguistic data, called bipolar complex fuzzy linguistic Aczel-Alsina power averaging operator, bipolar complex fuzzy linguistic Aczel-Alsina power weighted averaging operator, bipolar complex fuzzy linguistic Aczel-Alsina power geometric operator, and bipolar complex fuzzy linguistic Aczel-Alsina power weighted geometric operator with some dominant and fundamental laws such as idempotency, monotonicity, and boundedness. Moreover, we initiate the model of the Weighted Aggregates Sum Product Assessment technique with the help of consequent theory. In the context of geographic information systems and spatial information systems, coupling aims to find out the relationships among different components within a geographic information system, where coupling can occur at many stages, for instance, spatial coupling, data coupling, and functional coupling. To evaluate the above dilemma, we perform the model of multi-attribute decision-making for invented operators to compute the best technique for addressing geographic information systems. In the last, we deliberate some numerical examples for comparing the ranking results of proposed and prevailing techniques.